1 EDUCATION Revista Mexicana de Física E 59 (2013) JANUARY JUNE 2013 A handy exact solution for flow due to a stretching oundary with partial slip U. Filoello-Nino a, H. Vazquez-Leal a, Y. Khan, A. Perez-Sesma a, A. Diaz-Sanchez c, A. Herrera-May d, D. Pereyra-Diaz a, R. Castaneda-Sheissa, V.M. Jimenez-Fernandez a, and J. Cervantes-Perez a a University of Veracruz, Electronic Instrumentation and Atmospheric Sciences School, Cto. Gonzalo Aguirre Beltrán S/N, Zona Universitaria, Xalapa, Veracruz, México 91000, Department of Mathematics, Zhejiang University, Hangzhou , China. c National Institute for Astrophysics, Optics and Electronics, Electronics Department, Luis Enrique Erro #1, Tonantzintla, Puela, México. d Micro and Nanotechnology Research Center, University of Veracruz, Calzada Ruiz Cortines 455, Boca del Rio, Veracruz, Mexico, Received 27 Novemer 2012; accepted 21 March 2013 In this article we provide an exact solution to the nonlinear differential equation that descries the ehaviour of a flow due to a stretching flat oundary due to partial slip. For this, we take as a guide the search for an asymptotic solution of the aforementioned equation. Keywords: Nonlinear differential equations; partial slip; stretching oundary; non-newtonian fluids. PACS: d; x 1. Introduction The flow due to a stretching oundary is important in many engineering processes. For instance in the glass fire drawing, crystal growing, polymer industries , and extrusion processes for the production of plastic sheets [2,3,8,9,10], among many others. Unlike what happens with Newtonian fluids (water, mercury, glycerine, etc.) which usually use no slip oundary conditions,  (pages ), there are cases where partial slip etween the fluid and the moving surface may occur. Examples include emulsions, as mustard and paints; solutions of solids in liquids finely pulverized, such as the case of clay, and polymer solutions . In these cases the oundary conditions are adequately descried y Navier s condition, which states that the amount of relative slip is proportional to local shear stress . As aforementioned, in this study we provide an exact solution to the nonlinear differential equation that descries the ehaviour of a flow due to partial slip, and therefore a non Newtonian fluid should e involved. Nevertheless for the sake of simplicity, we will consider the adequate limit cases, in order to justify the use of Newtonian equations for fluids. For instance, the Newtonian approximation for Bingham plastics (like the clay) amounts to consider fluids, in the limit of small values of the so called, yield stress  (page. 233). On the other hand, for the case of pseudo-plastic non Newtonian fluids (generally, aqueous solutions of water solule polymers show this ehaviour), their Newtonian limit amounts to consider the limits for large values of shear rate and apparent viscosity constant  (page. 233). Although there are solutions to this prolem [8,9,10], the solving procedures are not easy to follow for undergraduates in physics, mathematics and engineering. Therefore, we propose a straightforward methodology, ased on elementary differential and integral calculus, employing as a guide the search for an asymptotic solution of the afore mentioned prolem. The systematic procedures used to determine qualitatively, the asymptotic ehaviour for solutions of a differential equation, elong to the qualitative theory of nonlinear differential equations  (page. 334). Unlike the aove, this study proposes an asymptotic analytical solution, which turns out to e the exact solution of the prolem. 2. Governing Equations Consider a two dimensional stretching oundary (see Fig. 1). Where the velocity of the oundary is approximately proportional to the distance X to the origin , so that U = X. (1) Let (u, v) e the fluid velocities in the (X, Y ) directions, respectively. In this case, the oundary conditions are adequately descried y Navier s condition which states that the amount of relative slip is proportional to local shear stress. u(x, 0) U = kν u (X, 0), (2) Y where k is a proportional constant and ν is the kinematic viscosity of the ulk fluid. The relevant equations for this case are Navier-Stokes and continuity uu X + vu Y + p X ρ ν(u XX + u Y Y ) = 0, (3) uv X + vv Y + p Y ρ ν(v XX + v Y Y ) = 0, (4) u X + v Y = 0, (5) where ρ and p are density and pressure, respectively.
2 52 U. FILOBELLO-NINO et al that is u X ν y (x), (8) where prime denotes differentiation with respect to x. Since y = y(x), after integrating (8), we otain u ν y (x)x, (9) (y choosing an aritrary function of x, zero). In order to simplify the equations of motion, we introduce the following constant of proportionality into (6) FIGURE 1. Schematic showing a stretching oundary. v = νy(x), (10) and, therefore, (9) is rewritten as u = y (x)x. (11) Thus, expressing velocity field according to (10), (11), and (7); (5) is automatically satisfied. Next, we will show that (3) adopts a simpler form under these assumptions. By using (7), (10), and (11), (3) can e rewritten as uy (x) + ν vxy (x) p x ρ 2 Xy (x) = 0, (12) FIGURE 2. Source in front of a wall. In order to satisfy the equation of incompressiility (5), we will motivate a transformation, which contains implicitly the functional form of the velocity field. We will see that in this fashion, the motion equations are reduced to a single ordinary nonlinear differential equation. To this end, consider the case of a source of intensity Q in front of a wall (axis X), as it is shown in Fig. 2 . By symmetry arguments, it is expected that streamlines pattern shown in Fig. 2 have, in general terms, a symmetry similar to those in Fig. 1 as a increases. From  it is possile to show that the value of v component, when a, is v(y ) = Y Q 2πa 2, noticing that v < 0 and it is only a function of Y. Using as a guide the aove argument, we define a function y(x), such that v y(x), (6) where y(x) 0, 0 x < and x = Y ν. (7) From (5) and (6) we deduce that u X = v Y y Y (x) = ν y (x), where we have employed the chain rule from differential calculus. It should e noted that p x = 0, since the fluid motion is caused y the fluid eing dragged along y the moving oundary, therefore uy (x) + ν vxy (x) 2 Xy (x) = 0, (13) sustituting (10) and (11) into (13), we otain y y 2 + yy = 0. (14) To deduce the oundary conditions of (14), we see that v(y = 0) = 0, (see Fig. 1); therefore, from (7) and (10), is clear that y(0) = 0, (15) in the same way, from the condition lim u(x, Y ) = 0, Y and (7) (see Fig. 1), we otain the following oundary condition y ( ) = 0. (16) To conclude, y sustituting (1) and (11) into the Navier s condition (2), we otain Xy (0) X = kν u (X, 0), (17) Y y the chain rule we rewrite (17) as ( ) ( u dx Xy (0) X = kν (X, 0) x dy ), (18)
3 A HANDY EXACT SOLUTION FOR FLOW DUE TO A STRETCHING BOUNDARY WITH PARTIAL SLIP 53 after sustituting (7) and (11) we get where we have defined y (0) = 1 + Ky (0), (19) K = k ν. (20) In the next section we will solve (14) with oundary conditions (15), (16), and (18). 3. The Exact solution of a two dimensional Viscous Flow Equation In order to otain an exact solution for (14) we take as a guide the search for an asymptotic solution of the same equation. Rewriting (14) in the form and defining it is possile to express (21) as and the derivative of (21) as follows y = y 2 y y, (21) y 1 = y, (22) y 1 y yy 1 = 0, (23) y 1 = y 1(2y 1 y 1 y 1 ) y 1 (y 2 1 y y ). (24) Equations (23) and (24) take the following limit forms when y 1 ( ) 0 (oundary condition (16)) y 1 + Ly 1 = 0, (25) y 2 1 y 1y 1 = 0, (26) (ecause the condition y 1 ( ) 0 implies that y( ) L, where L is constant. Also from Fig. 1 and (10), it follows that v < 0 and L > 0). Taking the square of (25) and sustituting into (26), we otain y 1 = L 2 y 1. (27) In order to solve (27), we propose the following change of variale z = y 1, (28) in such a way that (27) adopts the form Equation (29) has the known solution z = L 2 z. (29) z(x) = A exp(lx) + B exp( Lx), (30) where A and B are constants. To avoid that z, when x (from (16), (22), and (28) is clear that z 0 for that limit), we choose A = 0 so that (30) adopts the simpler form also, from (28) and (31), we otain z(x) = B exp( Lx), (31) y 1 = B exp( Lx), (32) therefore, after integrating the aove equation, we otain y 1 = B L exp( Lx) + c 1. (33) The condition y 1 ( ) = 0 (see (16) and (22)) leads to c 1 = 0, therefore after integrating (34), is otained y 1 = B exp( Lx), (34) L y(x) = B L 2 exp( Lx) + c 2. (35) Since y(0) = 0, then c 2 = (B/L 2 ), and (35) ecomes y(x) = B (exp( Lx) 1). (36) L2 Finally, the condition y( ) L is satisfied y choosing B = L 3, so that and y(x) = L(1 exp( Lx)). (37) On the other hand, from (37), we deduce that y (0) = L 2, (38) y (0) = L 3. (39) The sustitution of (38) and (39) into (19) leads to a general relation etween K and the asymptotic form of the solution given y y = L KL 3 + L 2 = 1. (40) For the case K = 0 [4,5], (40) and (37) adopt the form respectively. L = 1, (41) y(x) = 1 exp( x), 0 x, (42)
4 54 4. U. FILOBELLO-NINO et al Discussion The sustitution of (37) into (14) reveals that (37) is, indeed, an exact general solution not just an approximation; although our process was aimed to find an asymptotic solution that would satisfy oundary conditions (15), (16), and (19). As a matter of fact, for the particular case of inviscid flow K = 0 ((41) and (42)) was reported in [4,5] as a rare closed exact solution for (14). F IGURE 5. Streamlines for K = 0. F IGURE 3. Function y(x) for several values of K. It is noteworthy that (40) relates the asymptotic form of the solution y = L with the constant K, and the latter is related to the fluid viscosity (see (20)); so that, in principle, (40) determines in advance the asymptotic value of the solution, from the value of the viscosity. Similarly, (39) and (40) provide a general way to determine the value of y 00 (0) in terms of K. These values are often difficult to calculate and in the literature can e found tales that provide some values ut just for a few values of K . Figure 3 and Fig. 4 show functions y(x) and y 0 (x) respectively for various values of K; these functions determine the velocity field through (10) and (11). Figure 5 shows a sketch for several streamlines when K = Conclusion An important task is to find analytic expressions that provide a good description of the solution to the nonlinear differential equations like (14). For instance, the flow induced y a stretching sheet is adequately descried y (37) and (40). An important result for practical applications it follows that (40) relates the asymptotic form of the solution y = L, with the fluid viscosity, so that in principle (40) determines in advance the asymptotic value of the solution from the value of the viscosity. This work showed, y means of a simple procedure, that some nonlinear differential equations may e solved in exact form taking as a guide the search for an asymptotic solution of the same equation. Is clear that this procedure could e useful, at least, to find approximate solutions for some equations. F IGURE 4. Function y 0 (x) for several values of K. 1. W.F. Hughes and J.A. Brighton, Dina mica De Los Fluidos (Mc Graw Hill. 1967). 2. V. Aliakar, A. Alizadeh- Pahlavan and K. Sadaghy, Nonlinear Sciences and Numerical Simulation, Elsevier (2007) M.M. Rashidi and D.D. Gangi, J. Homotopy perturation method for solving flow in the extrusion processes.ije Transactions A: Basics 23 (2010) C.Y. Wang, Chemical Engineering Science 23 (2002) L.J. Crane, Flow past a stretching plate. Zeitschrift fuer Angewandte Mathematik und Physik 21 (1970) J. Vleggar, Chemical Engineering Science 32 (1977) B.R. Munson, D.F. Young and T.H. Okiishi, Fundamentals of Fluid Mechanics (John Wiley and Sons, Inc. Copyright 2002). 8. H.I. Anderson, Acta Mech. 158 (2002) C.Y. Wang, Nonlinear Anal. Real World Appl. 10 (2009)
5 A HANDY EXACT SOLUTION FOR FLOW DUE TO A STRETCHING BOUNDARY WITH PARTIAL SLIP T. Fang, J. Zhang, and S. Yao, Commun Nonlinear Sci. Numer. Simul. 14 (2009) F. Simmons, Ecuaciones Diferenciales con Aplicaciones y Notas Históricas (Mc Graw Hill. 1977).